
where its state equation is similar to the system (REF ), but instead of a known state constraint, an observation/measurement signal \(y_t \in \mathbb {R}^{n_{\color {black} {y}}}\)  is known/measured with a known observation function \({\mu }(\cdot ):\mathbb {R}^{n_x+n_u} \rightarrow \mathbb {R}^{n_{{\color {black} y}}} \) , known matrix \(V \in \mathbb {R}^{n_{{\color {black} y}} \times {\color {black} n}_v}\)  and measurement noise signal \(v_t \in [\underline{v},\overline{v}] \in \mathbb {IR}^{n_v}\) . It can be easily verified that the observation equation can be equivalently written as \({\nu }(x_t) \triangleq {\mu }(x_t,u_t) \in \mathcal {Y}_t \triangleq [y_t-\overline{s},y_t-\underline{s}]\) , where \(\overline{s}=V^{+}\overline{v}-V^{-}\underline{v}\) , \(\underline{s}=V^{+}\underline{v}-V^{-}\overline{v}\) , \(V^{+} \triangleq \max (V,\mathbf {0}_{n_v})\)  and \(V^{-} \triangleq V^{+} - V\) , with \(\mathbf {0}_{n_v}\)  being a zero vector in \(\mathbb {R}^{n_v}\)  [1]}. This transforms the system (REF ) into the form of (REF ) and hence, Proposition REF  directly applies, where \([\underline{x}^p_t,\overline{x}^p_t]\)  is the state interval from the prediction/propagation step and \([\underline{x}^u_t,\overline{x}^u_t]\)  is the updated state interval after a measurement update step. Further, if (REF ) is a sampled-data system, i.e., the system dynamics is continuous and the observations are sampled in discrete-time, our proposed approach still directly applies  with minor modifications.
